|
| And so, on the second day, the Vizier got 2 grains,
on the third, he got 4 grains, and the young prince couldn't help himself
making fun of the Vizier.
By the sixth day, the Vizier got 32 grains of wheat. By the eighth day
and the end of the first row, he got a mere 128 grains. By the sixteenth
day and the end of the second row, he got 32,768 grains.
Where was this leading to? Was it worth it for the Vizier?
By the end of the game (it was a mind game, wasn't it?) can you guess
how many grains the Vizier would get in all?
The prince could not provide enough grains to give the Vizier a chess
board's worth of grains. Why? Study the progress of the number of grains
on the individual squares and the corresponding running total in the table
below. There are not enough grains in the whole world to give the Vizier
the sum of 18,446,744,073,709,551,615 grains! Indeed, that many
grains would cover the entire earth several inches
deep. |
|
Square
|
Grains on Square
|
Running Total
|
|
1
|
1
|
1
|
|
2
|
2
|
3
|
|
3
|
4
|
7
|
|
4
|
8
|
15
|
|
5
|
16
|
31
|
|
6
|
32
|
63
|
|
7
|
64
|
127
|
|
8
|
128
|
255
|
|
9
|
256
|
511
|
|
10
|
512
|
1,023
|
|
11
|
1,024
|
2,047
|
|
12
|
2,048
|
4,095
|
|
13
|
4,096
|
8,191
|
|
14
|
8,192
|
16,383
|
|
15
|
16,384
|
32,767
|
|
16
|
32,768
|
65,535
|
|
17
|
65,536
|
131,071
|
|
18
|
131,072
|
262,143
|
|
19
|
262,144
|
524,287
|
|
20
|
524,288
|
1,048,575
|
|
21
|
1,048,576
|
2,097,151
|
|
22
|
2,087,152
|
4,194,303
|
|
23
|
2,097,152
|
8,388,607
|
|
24
|
8,388,608
|
16,777,215
|
|
25
|
16,777,216
|
33,554,431
|
|
26
|
33,554,432
|
67,108,863
|
|
27
|
67,108,864
|
134,217,727
|
|
28
|
134,217,728
|
268,435,455
|
|
29
|
268,435,456
|
536,870,911
|
|
30
|
536,870,912
|
1,073,741,823
|
|
31
|
1,073,741,824
|
2,147,483,647
|
|
32
|
2,147,483,648
|
4,294,967,295
|
|
33
|
4,294,967,296
|
8,589,934,591
|
|
34
|
8,589,934,592
|
17,179,869,183
|
|
35
|
17,179,869,184
|
34,359,738,367
|
|
36
|
34,359,738,368
|
68,719,476,735
|
|
37
|
68,719,476,736
|
137,438,953,471
|
|
38
|
137,438,953,472
|
274,877,906,943
|
|
39
|
274,877,906,944
|
549,755,813,887
|
|
40
|
549,755,813,888
|
1,099,511,627,775
|
|
41
|
1,099,511,627,776
|
2,199,023,255,551
|
|
42
|
2,199,023,255,552
|
4,398,046,511,103
|
|
43
|
4,398,046,511,104
|
8,796,093,022,207
|
|
44
|
8,796,093,022,208
|
17,592,186,044,415
|
|
45
|
17,592,186,044,416
|
35,184,372,088,831
|
|
46
|
35,184,372,088,832
|
70,368,744,177,663
|
|
47
|
70,368,744,177,664
|
140,737,488,355,327
|
|
48
|
140,737,488,355,328
|
281,474,976,710,655
|
|
49
|
281,474,976,710,656
|
562,949,953,421,311
|
|
50
|
562,949,953,421,312
|
1,125,899,906,842,623
|
|
51
|
1,125,899,906,842,624
|
2,251,799,813,685,247
|
|
52
|
2,251,799,813,685,248
|
4,503,599,627,370,495
|
|
53
|
4,503,599,627,370,496
|
9,007,199,254,740,991
|
|
54
|
9,007,199,254,740,992
|
18,014,398,509,481,983
|
|
55
|
18,014,398,509,481,984
|
36,028,797,018,963,967
|
|
56
|
36,028,797,018,963,968
|
72,057,594,037,927,935
|
|
57
|
72,057,594,037,927,936
|
144,115,188,075,855,871
|
|
58
|
144,115,188,075,855,872
|
288,230,376,151,711,743
|
|
59
|
288,230,376,151,711,744
|
576,460,752,303,423,487
|
|
60
|
576,460,752,303,423,488
|
1,152,921,504,606,846,975
|
|
61
|
1,152,921,504,606,846,976
|
2,305,843,009,213,693,951
|
|
62
|
2,305,843,009,213,693,952
|
4,611,686,018,427,387,903
|
|
63
|
4,611,686,018,427,387,904
|
9,223,372,036,854,775,807
|
|
64
|
9,223,372,036,854,775,808
|
18,446,744,073,709,551,615
|
|
